Topology Proceedings

Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124

ACTION OF CONVERGENCE GROUPS

NANDITA RATH

Abstract. This is a preliminary report on the continuous action of convergence groups on convergence spaces. In particular, the convergence structure on the homeomorphism group and its continuous action are investigated in this paper. Also, attempts have been made to establish a one-to-one correspondence between continuous action of a convergence group and its homeomorphic representation on a convergence space.

1. Introduction In algebra, a homomorphism of a group G into the symmetric group S(Ω) of all permutations on the phase space Ω is known as a permutation representation of G on Ω [5]. It is also known that there exists a one-to-one correspondence between the permutation representations of G on Ω and the actions of G on Ω. If Ω = X is a compact or a locally compact topological space and G is a topological group, then there is a one-to-one correspondence between continuous homomorphisms of a topological group G into the homeomorphism group H(X) and the continuous group actions of G on X [12]. However, for general topological spaces this one-to- one correspondence is not available, since in such case H(X) does not have a group topology which is admissible [11].

2000 Mathematics Subject Classiﬁcation. Primary 57E05, 57S05, 54A20; Secondary 22F05. Key words and phrases. Homeomorphism group, continuous group action, convergence space, limit space, pseudotopology. 601 602 NANDITA RATH

So in establishing such a one-to-one correspondence between continuous representations and continuous group actions of a topological group, the question of existence of an appropriate geometric structure on X which induces an admissible and compatible structure on H(X) is crucial. Recently, Di Concilio [4] has proved that a Tychonoﬀ space X, which is also rim-compact induces an admissible and compatible topological structure on H(X). This result can be further generalized by replacing the topology by a special type of convergence structure [10]. In Section 4 of this paper the author tries to investigate the nature of convergence structure on H(X) corresponding to the existing convergence structure on X. In Section 5 attempts have been made to establish a one-to-one correspondence between the homeomorphic representations and continuous group actions of a special type of convergence group, called limit group.

2. Preliminaries For basic definitions and terminologies related to filters the reader is referred to [14], though a few of the definitions and notations, which are frequently used will be mentioned here. Let F(X) denote the collection of all filters on X and P(X) denote all the subsets of X. If B is a base [14] of the filter , then is said to be generated by B and we write = [B]. ForF each x F X, x˙ = [ x ] is the fixed ultra filter containingF x . If and ∈ F(X){ and} F G = φ for all F , G {, then} F denotesG ∈ the filter generated∩ 6 by F G :∈F F and∈ GG F. If∨ GF and G such that F G{ =∩φ, then∈ we F say that ∈ G} fails∃ to∈ exist F . If (X,.∈ G) is a group, then∩ we can define a binaryF ∨G operation ‘ ’ on F(X) as 1 ◦ 1 = [ FG : F and G ], and − = [ F − : F ]. FGHowever,{ in general,∈ F (F(X), ∈) G is a monoid,F not{ a group∈ , since F} e˙ 1, where e is the identity◦ element in (X,.). ≥ FF − For a set X, consider q F(X) X. Conventionally, for ( , x) F(X) X, we write ⊆q x, and× say that ‘q convergesF to∈ x ’, whenever× ( , x) Fq. → F − F ∈ Definition 2.1. Let X and q be as above. For , F(X) and x X, consider the following conditions. F G ∈ ∈ q (c1)x ˙ x for each x X. → ∈ ACTION OF CONVERGENCE GROUPS 603

q q (c2) and x = x. F ≤q G F → q⇒ G → (c3) x = x˙ x. F →q ⇒ Fq ∩ → q (c4) x, G x = x. F → → ⇒ F ∩ G → q q (c5) If for each ultrafilter , x, then x. G ≥q F G → F → q (c6) For each x X, νq(x) x, where νq(x) = : x . ∈ → ∩{F F → } (c7) For each x X, and for each V νq(x), W νq(x) such ∈ ∈ ∃ ∈ that W V and y W V νq(y). ⊆ ∈ ⇒ ∈ The function q is called a preconvergence structure (respectively, convergence, limit, pseudotopology, pretopology, topology) if it satisfies (c1) (c2)(respectively, (c1) (c3), (c1) (c4), (c1) − − − − (c5), (c1) (c6), (c1) (c7)). It should be noted that all these ax- − − ioms are not independent. For example, (c1)and (c4) imply (c3) and (c2)and (c6) imply (c5).Convergence space, limit space, pseudotopological space, pretopological space and topological space are defined likewise. More details on convergence spaces and related topics can be found in [9]. Note that limit spaces used to be called convergence spaces by Binz [2], Park [10] and several contemporary topologists. For any x X, let q(x) = F(X): q x . ∈ {F ∈ F → } A convergence space (X, q) is said to be q T2 or Hausdorff iff x = y, whenever any filter x, y. · Compact iff each ultrafilter on X converges to at leastF → one point in ·X. A mapping f :(X, q) (Y, p) is continuous iff whenever q x, f( ) p f(x). Furthermore,→ a continuous mapping f F → F → 1 is a homeomorphism, if it is bijective and f − is also continuous. The set H(X) (respectively, C(X)) denotes the set of all self homeomorphisms (respectively, continuous functions) on X. H(X) forms a group with respect to composition of maps. Next, another important category of spaces is introduced which exists in literature mostly as a means to generalize the completion theory for uniform spaces [13]. For a detailed information on Cauchy spaces the reader is referred to [9]. Definition 2.2. Let C be a set of filters on a set X. The pair (X,C) is called a Cauchy space, if (1)x ˙ C, for each x X. (2) ∈ C and ∈implies C. (3) F, ∈ C andF ≤ G existsG imply ∈ C. F G ∈ F ∨G F ∩ G ∈ 604 NANDITA RATH

Associated with each Cauchy structure C on X, there is a qc convergence structure qc, which is defined as x, iff x˙ C. A Cauchy space is said to be completeF →, if every F ∩ ∈ C is qc-convergent. Note that every convergence space is notF ∈ a Cauchy space. A convergence structure q on X is said to be Cauchy compatible, if there exists a Cauchy structure on X such that q = qc. Lemma 2.3. [9] A convergence space (X, q) is Cauchy compatible if and only if it is a limit space and for any x = y X, either q(x) = q(y) or q(x) q(y) = φ.( ) 6 ∈ ∩ ∗ 3. Convergence Groups Convergence groups, especially limit groups have been studied in detail in the last half century. For clarity of notations and terminologies the reader is referred to [6, 8, 10]. Definition 3.1. A triplet (X, q, ) is a pre convergence group, if · − (cg1)(X, q) is a pre-convergence space (cg2)(X, ) is a group · q q 1 q 1 (cg3) x and y implies that − xy− . Note thatF → the binaryG → operation ‘.’ andFG the inverse→ operation in the group (X,.) are continuous with respect to the pre-convergence structure q. A pre-convergence group is a convergence group (respectively, limit group, pseudotopological group, pretopological group, topological group), iff (X, q) is a convergence space (respectively, limit space, pseudotopological space, pretopological space, topological space). Lemma 3.2. [8] If e denotes the identity element in this group, then xνq(e) = νq(e)x = νq(x), x X. ∀ ∈ Lemma 3.3. [8] The left translation x ax, for a fixed a G → 1 ∈ (respectively, the right translation x xa), the map x x− and 1→ → the inner automorphisms x axa− are all homeomorphisms of X onto X. → The proof of the lemma follows from (cg3). Note that every pre- convergence group is homogeneous. So some of the local properties can be proved at a single point. Lemma 3.4. The property ( ) in Lemma 2.3. always holds in a pre-convergence group (X, q, )∗. · ACTION OF CONVERGENCE GROUPS 605

Proof. Suppose q(x) q(y) = φ and let be such that q x and q ∩ 6 1 Fq F → y. Let q(x). Then, − e, where e is the identity F → G ∈ 1F Gq → q in X. This implies that − y, since y. Therefore =e ˙ q y. Similarly, weFF canG show → that q(y)F →q(x). G G → ⊆ Lemma 3.5. Let , F(X), where (X, ) is a group. Then exists impliesF thatG ∈ 1 . · F ∨ G F ∩ G ≥ FG− G Proof. Let F and G1,G2 . Since G1 G2 = φ, 1 ∈ F ∈ G ∩ 6 1 F FG− G2. Also, G exists implies that G2 FG− G2. ⊂ 1 F ∨ ⊂ 1 This proves the lemma. Lemma 3.6. [14] is an ultraﬁlter on X if and only if A B implies either A orLB is in . ∪ ∈ L L Lemma 3.7. [3] If (X, q, ) is a convergence group such that q is pretopological, then (X, q, )· is a topological group. ·

4. Homeomorphism Groups If (X, q) is a pre-convergence space, then the set of all self-homeomorphisms on X is denoted by H(X). Under the composition of maps H(X) forms a group, called the homeomorphism group of X. We can define a function σ on the filters on H(X) as follows: For any filter Φ on H(X) and f H(X), we define f σ(Φ) or equivalently, Φ σ f, iff ∈ ∈ (1) x X, → q x in X implies that Φ( ) q f(x) in X, ∀ ∈q F → 1 qF →1 (2) x in X implies that Φ− ( ) f − (x) in X, F →1 1 F → where Φ− = [ P − : P Φ ], and Φ( ) = w(Φ ) with w as the evaluation{ map, w ∈: H(}X) X F X. A pre-convergence× F structure on H(X) is said to be admissible,× → if the evaluation map is continuous. Note that if X is a locally compact (respectively, compact topological space), then σ coincides with the g topology [1] (respectively, γ topology [1]) . − − In 1972, Park [10] has shown that if q is a limit structure on X, then (H(X), σ) is a limit group and σ is the coarsest admissible limit structure on H(X). We intend to study the convergence structures on X which induce a suitable admissible convergence structure on the homeomorphism group H(X). The following corollary can be derived from the proof of Theorem 5 [10]. 606 NANDITA RATH

Corollary 4.1. If q is a pre-convergence structure on X, then H(X) is a pre-convergence group and σ is the coarsest admissible preconvergence structure on H(X). Proposition 4.2. If (X, q, ) is a convergence group, then σ is the coarsest admissible group convergence· structure on H(X). Proof of Proposition 4.2 follows from Lemma 3.5. Proposition 4.3. If q is a pseudotopology on X, then σ is the coarsest admissible pseudotopology on H(X). Proof. Since (X, q) is also a preconvergence space, it follows from Corollary 4.1 that H(X) satisfies (c1), (c2) of Definition 2.1 and (cg3) of Definition 3.1. So (H(X), σ, ) is a preconvergence group. So it is enough to show that σ is a pseudotopology◦ on H(X), whenever q is a pseudotopology on X. Let Φ be a filter on H(X) such that Φ is not σ -convergent, but for any ultrafilter Ψ Φ, Ψ σ f for some f H(X). Since Φ 9σ f, there exists a filter≥ →q x in X such that∈ Φ( ) 9q f(x). Since q is a pseudotopology,F → there exists an ultrafilterF Φ( ) such that 9q f(x). < ≥ F < Claim: If is an ultrafilter on X such that Φ( ), then an ultrafilter < Φ such that ( ). < ≥ F ∃ L ≥ < ≥ L F Proof of the claim: Let Θ = Λ : Λ is a filter on H(X) with Λ Φ and Λ( ) . A standard{ Zorn’s lemma argument shows that≥ Θ has a< maximal ≥ F } element. Let be the maximal element in Θ. So it is enough to show that is anL ultrafilter. Let A B , we need to show that either A orLB is in . If neither A nor∪ B∈is L in , then L L A = [ A M : M ] , B = [ B M : M ] . L { ∩ ∈ L} L L { ∩ ∈ L} L Since is maximal in Θ, A, B both fail to be in Θ. So L L L there exist M1,M2 and there exist F1,F2 such that ∈ L ∈ F (M1 A)(F1), (M2 B)(F2) / . Let M = M1 M2 and F = ∩ ∩ ∈ < ∩ F1 F2. Since A B , (M (A B))(F ) ( ), which implies∩ that (M ∪(A B∈))( LF ) ∩. However,∪ (M ∈(A L FB))(F ) ∩ ∪ ∈ < ∩ ∪ ⊆ (M1 A)(F1) (M2 B)(F2) / , since is an ultrafilter. So (M ∩(A B))(∪F ) / ∩, which leads∈ < to a contradiction.< This proves the∩ claim.∪ ∈ < Since is an ultrafilter Φ, σ f and therefore ( ) qLf(x). This implies that≥ Lq f →(x), which leads to a Lcontradiction.F → < → This proves Proposition 4.3. ACTION OF CONVERGENCE GROUPS 607

Note that in the special case when X is a locally compact T2 topological space, H(X) is a topological group and σ coincides with the g topology [1]. − The proof of the following proposition follows immediately from Theorem 5 [10], Lemma 2.3, Lemma 3.4 and Corollary 4.1. Proposition 4.4. If (X, q, ) is a limit group, then H(X) is a complete Cauchy space. · The Cauchy structure on H(X) can be explicitly given as Cσ = all σ convergent ﬁlters on H(X). −

5. Continuous Group Action Consider a function on X G into X, denoted by (x, g) xg. For any F X , g G, F g = ×f g : f F and for any A →G, F A = F g :⊆g A . ∈If F(X{) and g∈ G,} then g = [ F g⊆: F ], and∪{ for any∈ subset} FA ∈ G, A = [ ∈F A : F F ]. If {κ F(G∈),then F} κ = [ F A : F ,A⊆ κ F]. { ∈ F} ∈ F { ∈ F ∈ } Definition 5.1. Let (G, Λ,.) be a convergence group and (X, q) be a convergence space. G is said to act continuously on X, if the function X G X, denoted by (x, g) xg, x Xand g G, satisfies the× following→ conditions : → ∀ ∈ ∀ ∈ (a1) q x and κ Λ g, κ q xg, where F(X) and κ F(G∀). F → ∀ → F → F ∈ ∈(a2) xe = x, x X. (a3) (xg)h =∀xgh∈, x X and g, h G. ∀ ∈ ∀ ∈ Note that condition (a1) implies the continuity of the group action at each x X and g G. The action is said to be effective, if xg = x, x X∈ implies that∈ g = e. The action is free, if for some x, xg =∀x, ∈ g = e. An action of a group G is transitive on X, if for any x, y⇔ X, g G such that xg = y. ∈ ∃ ∈ Lemma 5.2. The conditions (a2) and (a3) are equivalent to (a20) e = and (a3 )( g)h = gh, F(X) and g, h G. F F 0 F F ∀F ∈ ∀ ∈ This can be verified by substitutingx ˙ for in (a20) and (a30). So it follows that if group G algebraically actsF on a set X, then G algebraically acts on the set F(X) of all filters on X, the action being defined as ( , g) g on F(X). F 7−→ F 608 NANDITA RATH

Example 5.3. The homeomorphism group H(X) acts continuously on (X, q), the action being defined by the map (x, f) xf = f 1(x), for all x X and f H(X). 7→ − ∈ ∈ Example 5.4. Any convergence group (G, Λ,.) acts on itself continuously, the action being defined by the right multiplication, x (a, x) a = ax, a, x G. This follows from (cg3) and (a1). The left7→ multiplication∀ (a,∈ x) = xa, a, x G is not an action, but (a, x) ax = x 1a, a, x G is an∀ action∈ of G on itself. In case 7→ − ∀ ∈ of right regular representation, if a1, a2 G, then there exists x = 1 x ∈ a1− a2 G such that a1 = a2 . Similarly, in the latter action y ∈ 1 a1 = a2, by choosing y = a1 a2− . So both these actions are tran- x 1 sitive. G also acts on itself by conjugation, i.e., a = x− ax. But this is not transitive in general. Next, we try to construct new group actions from a given group action of a convergence group G on a convergence space X. Example 5.5. If G acts continuously on X then G acts contin- g uously on Y = X X, where the action is defined by (x1, x2) = g g × (x1, x2), g G and x1, x2 X. Here it is assumed that Y has the product∀ convergence∈ structure∈ [8]. Proposition 5.6. Let (X, q) and (Y, p) be two limit spaces. If G acts continuously on (X, q) then G acts continuously on C(X,Y ) = f : f : X Y, f is continuous , with respect to the continuous {convergence→ structure Λ [2] on C(}X,Y ). 1 Proof. Define the action as (f, g) f g, where f g(x) = f(xg− ), x X and g G. Since for any7−→ q x in X, f( ) p f(x) ∀ ∈ ∈ 1 F → F → 1 and f g( ) = f( g− ), it follows from (a1) that f g( ) p f(xg− ), which showsF f g F C(X,Y ). Since f e(x) = f(xe) =Ff(→x), x X, ∈ 1 1 1 ∀ ∈1 f e = f and (f g)h(x) = f g(xh− ) = f((xh− )g− ) = f(x(gh)− ) = f gh(x) x X (f g)h = f gh. This proves (a2) and (a3). To prove ∀ ∈ Λ ⇒ γ 1 γ 1 (a1) let Φ f in C(X,Y ) and g in G. Then − g− in G and by→ the continuous action< of →G on X, for any< →q x in X, 1 q g 1 1 F →p g 1 <− x − in X. So, by definition of Λ, Φ( <− ) f(x − ), F → p g Fq → in Y, which implies that Φ<( ) f (x), x and therefore, Φ Λ f g. This proves PropositionF → 5.6. ∀F → < → In particular, G acts continuously on C(X), the space of self- continuous maps on X,whenever G acts continuously on (X, q). ACTION OF CONVERGENCE GROUPS 609

Proposition 5.7. Let (X, q) and (Y, p) be two limit spaces. If G acts continuously on (Y, p) then G acts continuously on C(X,Y ) = f : f : X Y, f is continuous , with respect to the continuous {convergence→ structure Λ [2] on C(}X,Y ). Proof. Define the action as (f, g) f g, where f g(x) = (f(x))g, x X and g G. Since f e(x) =7−→ (f(x))e = f(x) and (f g)h(x) = ∀(f(x∈))gh, x ∈X, (a2),(a3) are satisfied. Also for any q x, since f( ∀) ∈p f(x) and f g( ) = (f( ))g, it follows fromF (a1) → that f g( ) Fp (→f(x))g, which showsF f g F C(X,Y ). The continuity of theF action→ follows from the admissibility∈ of the continuous convergence structure and (a1) in Definition 5.1 by arguments similar to those of Proposition 5.6. Proposition 5.8. If (G , γ, )acts on the limit space (X, q), then G acts on (H(X), σ), the group· of self-homeomorphisms with respect to the double convergence σ. Proof. For any f H(X) and g G define the action as ∈ 1 ∈ (f, g) f g, where f g(x) = f(xg− ), x X . We first show 7−→g ∀ ∈g g that f H(X) for any f H(X). Let f (x1) = f (x2), i.e., 1 1 1 1 g− ∈ g− ∈ g− g− f(x1 ) = f(x2 ). This implies that x1 = x2 , since f is bijec- g tive and therefore x1 = x2. Next, for any y X, x X, such that 1 ∈ ∃ ∈ f g(xg) = f((xg)g− ) = f(x) = y, this implies that f g is bijective. The continuity of f g can be proved by following the same line of g 1 1 g argument as in Proposition 5.6. Define (f )− (x) = (f − (x)) , for all x X. Note that ∈ g 1 g 1 g g 1 g 1 g (f )− (f (x)) = (f − (f (x)) = (f − (f(x − )) = x and g g 1 1 g g 1 1 f ((f )− (x)) = f(((f − (x)) ) − ) = f(f − (x)) = x. g 1 q To show that (f )− is continuous, let x in X. Then 1 q 1 F1 → g q 1 g f − ( ) f − (x), which implies that (f − ( )) (f − (x)) , F →1 F → since f − is continuous and G acts continuously on X. So, g 1 q g 1 (f )− ( ) (f )− (x). Now,F we→ are going to show that the action on H(X) is continuous. Let Φ σ f in H(X) and γ g in G, then by arguments similar → < → q g to those of Proposition 5.6, we can show that Φ<( ) f (x) q 1 1 F → for all x in X. Also, (Φ<)− ( ) = (Φ− ( ))< and since 1 F →q 1 1 Fq 1 Fg Φ− ( ) f − (x) , (Φ− ( ))< (f − (x)) . So we have (Φ )F1( →) q (f g) 1(x), whichF implies→ that Φ σ f g. < − F → − < → 610 NANDITA RATH

Next, a one-to-one correspondence has been established between the homeomorphic representations of a convergence group and its continuous actions on a limit space. Lemma 5.9. If a convergence group G acts continuously on a convergence space X, then for each g G, the mapping g¯ : X X, defined by (x)¯g = xg is a homeomorphism.∈ → 1 1 Proof. It is routine to show thatg ¯ is a bijection andg ¯ − = g− . The fact thatg ¯ is continuous on X for each g G, follows from 1 ∈ g 1 1 (a1), if we substituteg ˙ for κ. Since (x)¯g − = (x) − , g¯ − is also continuous. This proves the lemma. We have seen that (H(X), σ) the homeomorphism group of X acts continuously on X . But every convergence group G acting continuously on X is not necessarily a homeomorphism group, though it is very close to a homeomorphism group when X is a limit space. Define a mapping ρ : G H(X) by ρ(g) =g, ¯ g G. Let ρ(A) = A¯ = g¯ : g A and for any→ filter κ on G, κ¯ = ρ∀(κ∈) = [ K¯ : K κ ]. { ∈ } { ∈ } Proposition 5.10. Let (X, ) be a limit space and (G, Λ,.) be a convergence group which acts= continuously on X. Then, the map ρ is a continuous group homomorphism.

Proof. Since gh =g ¯ h,¯ ρ is a group homomorphism. To prove the continuity of ρ, we consider the limit structure σ on H(X). Let g in G and κ Λ g in G. We need to show that ρ(κ) σ g.¯ → g→ Let = x in X. It suﬃces to show thatκ ¯( ) = x , and 1 1 F → g− 1 1 F → Λ κ¯− ( ) = x , becauseκ ¯− = κ− . Since = x and κ g, F → κ F → Λ→ by (a1) in Deﬁnition 5.1,κ ¯( ) = = x.¯ Since κ g in 1 Λ 1 F F → → G, by (cg3), κ− g− . Applying (a1) again we can show that 1 → κ¯ 1( ) xg− . This proves Proposition 5.10. − F →= The mapping ρ may be called the homeomorphic representation of the convergence group G on the limit space X. So every continuous group action on X is associated with a homeomorphic representation. If the continuous homomorphism ρ is one to one, i.e., Kernelρ = e , then it can be easily checked that G ρ(G). { } ≈ Next, we show that any continuous group homomorphism θ : G H(X) on G induces a continuous group action of G on X. → ACTION OF CONVERGENCE GROUPS 611

Proposition 5.11. Let θ : G H(X), be a continuous homomorphism. Then, G acts continuously→ on X. Proof. Let us deﬁne a function on X G X by (x, g) θ(g 1) θ(g 1) (θ(g)) 1 × 1 → 7→ x − , where x − = x − = (θ(g))− (x). Since θ is a ho- θ(e 1) momorphism, θ(e) = Id(X), and θ(gh) = θ(g)θ(h). So x − 1 θ((gh) 1) θ(h 1g 1) θ(h 1)θ(g 1) =θ(e− )(x) = x and x − = x − − = x − − = 1 1 1 θ(g 1) θ(g 1) θ(h 1) θ(h− )θ(g− )(x) = θ(h− )(x − ) = (x − ) − which imply q that (a2), (a3) hold. So it remains to prove (a1). Let x and κ Λ g, κ q xg, where F(X) and κ FF(G →). Then, ∀ κ → θ(κ 1F) → 1 qFθ ∈(g 1) ∈ = − = (θ(κ))− ( ) x − , since θ is continuous and theF evaluationF map is continuousF → with respect to σ on H(X). This proves Proposition 5.11. In Proposition 5.10 and 5.11, a one-to-one correspondence between homeomorphic representations of a convergence group G and its continuous action on a limit space X has been established. The question whether such a one-to-one correspondence exists for topological groups acting on general topological spaces still remains as an open question. Also, the transitive actions of a convergence group and actions of the quotient convergence group [7] are yet to be investigated.

References [1] R.F. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68 (1946), 593-610. [2] E. Binz, Continuous convergence on C(X), Lecture Notes in Mathematics 469 (1939), Springer-Verlag. [3] E. Cech, Topological spaces, Praha, 1966. [4] A. Di Concilio, Topologising homeomorphism groups of, rim-compact spaces, 16th Summer Topology Conference, New York (2001). [5] J.D. Dixon, B. Mortimer, Permutation groups, Springer-Verlag Publ., 1996. [6] R. Fric, On convergence spaces and groups, Math. Slovaca 27(4) (1977), 323-330. [7] K.H. Hofmann, S.A. Morris, The structure of compact groups, de Gruyter Studies in Mathematics 25, Walter de Gruyeter &Co (1998). [8] D.C. Kent, On convergence groups and convergence uniformities, Funda- menta Mathematicae (1967), 213-232. [9] D.C. Kent, N. Rath, Filter Spaces, Applied Categorical Structures 1(3) (1993), 297-309. [10] W.R. Park., Convergence structures on homeomorphism groups, Math. Ann. 199 (1972), 45-54. 612 NANDITA RATH

[11] W.R. Park., A note on the homeomorphism group of the rational numbers, Proc. Amer. Math. Soc. 42(2) (1974), 625-626. [12] V. Pestov, Topological groups: where to from here, Topology Proceedings 24 (1999), 421-502. [13] E.E. Reed, Completion of uniform convergence spaces, Math. Ann. 194 (1971), 83-108. [14] S. Willard, General topology, Addison-Wesley Publ., 1970.

Department of Mathematics and Statistics, University of Western Australia, Nedlands, WA 6907. E-mail address: [email protected]